Local Conformal Structure of Liouville Quantum Gravity

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Antti Kupiainen, Rémi Rhodes, Vincent Vargas

To cite this version:

Antti Kupiainen, Rémi Rhodes, Vincent Vargas. Local Conformal Structure of Liouville Quantum

Gravity. Communications in Mathematical Physics, Springer Verlag, 2018. �hal-01587078�

ANTTI KUPIAINEN¹, R´EMI RHODES², AND VINCENT VARGAS²

Abstract. Liouville Conformal Field Theory (LCFT) is an essential building block of Polyakov’s formu- lation of non critical string theory. Moreover, scaling limits of statistical mechanics models on planar maps are believed by physicists to be described by LCFT. A rigorous probabilistic formulation of LCFT based on a path integral formulation was recently given by the present authors and F. David in [14]. In the present work, we prove the validity of the conformal Ward identities and the Belavin-Polyakov-Zamolodchikov (BPZ) differential equations (of order 2) for the correlation functions of LCFT. This initiates the program started in the seminal work of Belavin-Polyakov-Zamolodchikov [4] in a probabilistic setup for a non-trivial Conformal Field Theory. We also prove several celebrated results on LCFT, in particular an explicit formula for the 4 point correlation functions (with insertion of a second order degenerate field) leading to a rigorous proof of a non trivial functional relation on the 3 point structure constants derived earlier in the physics literature by Teschner [42]. The proofs are based on exact identities which rely on the underlying Gauss- ian structure of LCFT combined with estimates from the theory of critical Gaussian Multiplicative Chaos and a careful analysis of singular integrals (Beurling transforms and generalizations). As a by-product, we give bounds on the correlation functions when two points collide making rigorous certain predictions from physics on the so-called “operator product expansion” of LCFT.

Contents

1. Introduction 2

1.1. Perspectives 5

1.2. History on LCFT and probabilistic approaches to CFT 6

1.3. Organization of the paper 7

2. Main results 7

2.1. Background and notations 7

2.2. Ward and BPZ identities 10

2.3. Relations on the 3 point structure constant 11

3. Properties of Liouville Correlation Functions 12

3.1. Integrability Properties 12

3.2. Integration by Parts Formula and a KPZ identity 12

3.3. Differentiability of Correlation Functions 14

4. Proof of the Ward and BPZ Identities 18

4.1. Proof of the first Ward Identity 18

4.2. Proof of the Second Ward Identity 21

4.3. Holomorphic identity for BPZ 26

4.4. Proof of Theorem 2.2 27

4.5. Proof of Theorem 2.3 29

5. Estimates on the correlation functions 31

5.1. Fusion Estimates 31

5.2. Proof of Proposition 3.1 33

5.3. H¨ older estimates 34

5.4. Proof of Lemma 4.2. 35

5.5. Proof of Lemma 4.3. 35

1Supported by the Academy of Finland,²Research supported in part by ANR grant Liouville (ANR-15-CE40-0013) Key words and phrases. Liouville Quantum Gravity, quantum field theory, Gaussian multiplicative chaos, Ward identities, BPZ equations.

1

6. Proof of Fusion Estimates 36

6.1. Regularized correlations and Proof of Lemma 3.2 36

6.2. Radial decomposition of the chaos measure 37

6.3. On the supremum of a drifted Brownian motion 38

6.4. Main technical lemma 39

6.5. Proof of Proposition 5.1 40

6.6. Proof of Proposition 5.3 41

6.7. Proof of Propositions 5.7 and 5.8 42

6.8. Proof of Proposition 5.5 43

6.9. Proof of Proposition 5.6 43

7. Appendix 44

7.1. Proof of Lemma 4.4 44

7.2. Proof of (4.34) 45

7.3. A special function entering the DOZZ formula 46

References 46

1. Introduction

Ever since the ground-breaking work of Belavin, Polyakov and Zamolodchikov (BPZ) in 1984 [4], the precise mathematical structure of local conformal symmetry uncovered in that work has been a challenge to mathematicians. The Conformal Field Theories (CFT) studied in [4] are believed to be limits of prob- abilistic objects, namely scaling limits of Gibbs measures of statistical mechanics models defined on two dimensional grids (or graphs). However, the full continuum formalism of CFT in the sense of BPZ has proven to be mathematically elusive except for a few cases among which the Gaussian Free Field (GFF) [26] and (partially) the Ising model at critical temperature (see [7, 8, 11, 13, 20, 25] for the latest developments on this).

One of the most intriguing CFT’s is the Liouville CFT (LCFT hereafter

¹

). It first appeared (in the context of String Theory) in Polyakov’s Liouville quantum gravity theory [33] of summation of random metrics and then in the 1988 work of Knizhnik, Polyakov and Zamolodchikov (KPZ) [27] on the relations between CFT’s on deterministic and random surfaces or in other terms on the relationship between statistical mechanics models on fixed grids and on random grids (random planar maps). An exact mathematical formulation of the so-called KPZ relation that appears in [27] is given in [15].

KPZ viewed random surfaces as a two dimensional manifold Σ equipped with a random Riemannian metric G whose law should be invariant under the group of diffeomorphisms Diff(Σ) of the surface. Guided by the fact that the space of smooth metrics on Σ is obtained as Diff(Σ) orbits of metrics of the form e

^σ

G ˆ where σ is a real valued function on Σ called conformal factor and ˆ G belongs to a finite dimensional moduli space of metrics they ended up looking for a law for the random field σ. They argued that the law of σ is described by LCFT. In what follows, we will only consider the case of the Riemann sphere Σ = ˆ C = C ∪{∞}

(one could also consider other topologies like the disk where one must also take into account non trivial specific issues linked to the presence of a boundary). In the case of the Riemann sphere, the metric is written as e

^γφ(z)

| dz |

²

and the law of φ is

(1.1) ν(dφ) = e

^−SL(φ)

ν

0

(dφ)

where S

L

is the Liouville Action functional

(1.2) S

L

(φ) =

¹_π

Z

C

( | ∂

z

φ(z) |

²

+ πµe

^γφ(z)

)d

²

z

1In our previous works, we also used the terminology Liouville quantum field theory with associated abbreviation LQFT.

Both terminologies (and theories) are completely equivalent since LQFT is in fact a CFT; we have decided to the use the abbreviation LCFT in this article to stress that it is a CFT.

with d

²

z the standard Lebesgue measure and ν

0

a putative ”flat” measure on some space of maps φ : C → R (in fact, there is an infinite constant hidden behind expression (1.2) that we omit in the introduction for the sake of clarity: see Section 2.1 for the precise formulation). One can notice that LCFT has two parameters γ and µ. According to KPZ [27], the parameter γ is determined by the particular CFT (e.g. model of statistical mechanics) which lives on the random surface. Such a CFT is characterized by its central charge c

M

(the M in the notation stands for matter since in Liouville quantum gravity this CFT is called a matter field) and the relation between γ and c

M

is c

M

= 25 − 6Q

²

where

(1.3) Q = 2

γ + γ 2 .

²

For instance, uniform random planar maps correspond to a CFT with central charge c

M

= 0 and therefore to γ = p

8/3, the Ising model on random planar maps corresponds to c

M

=

¹₂

hence to γ = √

3 and the GFF on random planar maps corresponds to c

M

= 1 hence to γ = 2. The parameter µ > 0 is called the cosmological constant and it makes the law (1.2) non Gaussian. It turns out that various quantities in LCFT have a simple scaling behaviour in µ (called “KPZ-scaling” in the physics literature) but we may not take µ to zero in LCFT.

LCFT is supposed to be as its name suggests also a Conformal Field Theory (CFT). Recall that this means in particular that there should exist primary conformal fields V

α

(z) defined for z ∈ C , i.e. random fields whose expectations in the Liouville law (1.1) are conformal tensors. More precisely, if z

1

, · · · , z

N

are N distinct points in C then for a M¨ obius map ψ(z) =

^az+b_cz+d

(with a, b, c, d ∈ C and ad − bc = 1)

(1.4) h

Y

N k=1

V

αk

(ψ(z

k

)) i = Y

N k=1

| ψ

^′

(z

k

) |

^−2∆αk

h Y

N k=1

V

αk

(z

k

) i

where we use the physicists’ notation h·i for the average with respect to the measure ν. The exponents ∆

αk

are called conformal weights. In LCFT the primary fields are called vertex operators in the physics jargon and are given by

(1.5) V

α

(z) = e

^αφ(z)

for suitable α ∈ C . As φ turns out to be a distribution valued random field, this definition requires a regularization and renormalization procedure (Section 2.1). In LCFT, the correlations in (1.4) were defined in [14] for N > 3 and under certain assumptions on the (α

k

)

16k6N

called the Seiberg bounds:

(1.6)

X

N k=1

α

k

> 2Q, ∀ k, α

k

< Q

The crucial property of a CFT with central charge c is however not the global conformal transformation property (1.4) but rather local conformal invariance. To define this in the probabilistic setup requires considering variation of the measure ν under a change of the geometry of the surface where the fields are defined. In general in local Quantum Field Theory one expects that such a variation is encoded in a random field, the stress-energy tensor (abbreviated SE tensor hereafter). Briefly, spelling this out in our 2d set-up, suppose the measure ν of the CFT can be defined in a set-up where the surface carries a smooth Riemannian metric G = P

2

i,j=1

g

ij

dx

ⁱ

⊗ dx

^j

; in this context, we will denote by h·i

G

averages with respect to the CFT in the background metric G. Let g

^ij

be the inverse matrix P

2

j=1

g

^ij

g

jk

= δ

_kⁱ

. Consider a one parameter family G

ǫ

where g

^ij_ǫ

= g

^ij

+ ǫf

^ij

whith f a smooth function with support in C \ ∪

^Nk=1

z

k

. Then one expects (1.7)

_dǫ^d

|

^ǫ=0

h

Y

N k=1

V

αk

(z

k

) i

^Gǫ

= X

2 i,j=1

1 4π

Z

C

f

^ij

(z) h T

ij

(z) Y

N k=1

V

αk

(z

k

) i

^G

vol

G

(d

²

z).

where vol

G

(d

²

z) is the volume form of G and T

ij

is by definition the SE tensor. In CFT, the SE tensor has two nontrivial components: in the complex coordinates they are T (z) := T

zz

(z) and ¯ T(z) := T

z¯¯z

(z).

2The central chargec_M of the CFT living on the random surface is not to be confused with the central chargec_Lof LCFT with parameterγ, which is also a CFT. These two CFTs are coupled independently withcL= 1 + 6Q²: equivalently, one has the relationcM+cL−26 = 0 discovered by Polyakov in [33].

Then, BPZ argued that T (z) encodes local conformal symmetries through the Conformal Ward Identities.

The first Ward identity says that the correlation function is meromorphic in the argument z of T (z) with prescribed singularities:

(1.8) h T (z) Y

N k=1

V

αk

(z

k

) i = X

N i=1

∆

αi

(z − z

i

)

²

h Y

N k=1

V

αk

(z

k

) i + X

N i=1

1 z − z

i

∂

zi

h Y

N k=1

V

αk

(z

k

) i .

Note in particular that the T insertion is holomorphic. The second Ward identity controls the singularity when two T -insertions come close

h T (z)T (z

^′

) Y

N k=1

V

αk

(z

k

) i = 1 2

c (z − z

^′

)

⁴

h

Y

N k=1

V

αk

(z

k

) i + 2

(z − z

^′

)

²

h T (z

^′

) Y

N k=1

V

αk

(z

k

) i

+ 1

z − z

^′

∂

z^′

h T (z

^′

) Y

N k=1

V

αk

(z

k

) i + . . . (1.9)

where the dots refer to terms that are bounded as z → z

^′

. Recall that here c is the central charge of the CFT. ¯ T(z) satisfies the complex conjugate identities.

As stressed by BPZ, these Ward identities are the fundamental structural property of a CFT and they are the starting point in showing that the CFT gives rise to a representation of the Virasoro Algebra with central charge c. In this paper, we define the SE tensor rigorously for LCFT and prove the Ward identities: see Theorem 2.1

³

. As an output of these Ward identities, we will adress the construction of the representation of the Virasoro Algebra in a forthcoming work.

The second set of fundamental identities for a CFT uncovered by BPZ goes under the name BPZ- equations. These are differential equations for correlation functions of the CFT that can be used to actually determine some of the correlation functions. BPZ uncovered a set of degenerate fields whose insertions to correlation functions lead to differential relations as in the case of Ward identities. In LCFT the two simplest degenerate fields are given by the vertex operators V

−^γ₂

and V

₋²

γ

: we will show that they satisfy the following second order linear differential equations:

( 1 α

²

∂

_z²

+

X

N i=1

∆

αi

(z − z

i

)

²

+ X

N i=1

1

z − z

i

∂

zi

) h V

α

(z) Y

N k=1

V

αk

(z

k

) i = 0.

(1.10)

where α = −

^γ₂

and α = −

²_γ

respectively.

Furthermore, we prove that equation (1.10) can be uniquely solved for N = 3 in terms of hypergeometric functions and the so-called three point structure constants of LCFT: this is because the global conformal invariance property (1.4) enables one to map equation (1.10) to a standard hypergeometric partial differential equation when N = 3. Let us mention that a key point in our analysis of equation (1.10) for N = 3 is the novel observation that assuming global conformal invariance (1.4) the real valued solution space to (1.10) is in fact (for most values of γ and the (∆

αi

)

16i6N

) a real one dimensional space: see lemma 4.4 in the Appendix for a precise statement on the hypergeometric partial differential equation. Now, one can notice that in a CFT the global conformal symmetry (1.4) fixes the three point correlation functions up to some fundamental constant C

γ

(α

1

, α

2

, α

3

): more precisely, one has

h Y

3 k=1

V

αk

(z

k

) i = | z

1

− z

2

|

^2∆12

| z

2

− z

3

|

^2∆23

| z

1

− z

3

|

^2∆13

C

γ

(α

1

, α

2

, α

3

)

with ∆

12

= ∆

α3

− ∆

α1

− ∆

α2

, etc. These constants C

γ

(α

1

, α

2

, α

3

) are called the three point structure constants and are building blocks of LCFT in the so-called conformal bootstrap approach. The boot- strap approach (see the review [34]) aims to give a construction of LCFT based on an exact formula for C

γ

(α

1

, α

2

, α

3

), the celebrated DOZZ-formula (after Dorn-Otto-Zamolodchikov-Zamolodchikov [16, 45])

3In the context of LCFT, the SE tensor defined formally by (2.16) below is nothing but the quantum analog of the classical SE tensorT(φ⋆) =−(∂zφ⋆)²+²_γ∂²_zzφ⋆ whereφ⋆minimizes the Liouville actionS_Lgiven by (1.2). The classical SE tensor was introduced more than a century ago by Poincar´e in his theory of the unifomisation of Riemann surfaces: one can read the nice introduction of [41] on this point.

and certain recursive rules to deduce from the DOZZ formula the n-point correlations for n > 3.

⁴

Using the solution of (1.10) and setting l(x) = Γ(x)/Γ(1 − x) we prove for α

1

, α

2

, α

3

satisfying the Seiberg bounds (1.6) that

(1.11) C

γ

(α

1

+

^γ₂

, α

2

, α

3

) C

γ

(α

1

−

^γ₂

, α

2

, α

3

) = − 1

πµ

l( −

^γ₄²

)l(

^γα₂¹

)l(

^α1₂^γ

−

^γ₄²

)l(

^γ₄

(¯ α − 2α

1

−

^γ₂

)) l(

^γ₄

(¯ α −

^γ₂

− 2Q))l(

^γ₄

(¯ α − 2α

3

−

^γ₂

))l(

^γ₄

(¯ α − 2α

2

−

^γ₂

)) .

where ¯ α = α

1

+ α

2

+ α

3

. This is the content of Corollary 2.5. Relation (1.11) was obtained earlier in the physics literature by Teschner [42] using clever but non rigorous conformal bootstrap techniques. Relation (1.11) is a major first step in a rigorous proof of the DOZZ formula: see next subsection.

Finally, the correlation functions of primary fields in CFT are singular as two points z

i

, z

j

get together:

h Y

N k=1

V

αk

(z

k

) i ∼ | z

i

− z

j

|

^−ηij

(1.12)

with η

ij

∈ R and ∼ denotes equivalence (up to logarithmic corrections) as | z

i

− z

j

| → 0. The study of these singularities is important as it gives information about the fusion rules, another of the structures of CFT uncovered by BPZ. More generally, the so-called operator product expansion in the physics literature corresponds to studying equivalence (1.12) at order 1 and higher in | z

i

− z

j

| . In Liouville theory, the operator product expansion is quite subtle and it is expected that the asymptotics in (1.12) have logarithmic corrections in | z

i

− z

j

|

⁵

. In this paper, as technical building blocks in the proof of the Ward identities and the BPZ equations, we prove detailed estimates on these singularities and in particular we prove that the logarithmic corrections indeed are present: see section 5 where are stated general fusion estimates.

Let us next describe briefly the mathematical content of the paper. The Liouville term R

C

e

^γφ(z)

d

²

z in the action functional (1.2) is an example of Gaussian Multiplicative Chaos (GMC), a random multifractal measure on C . The Liouville correlation functions turn out to be nonlinear functionals of the GMC measure and the study of their regularity boils down to a careful analysis of the GMC measure. In particular a variation of the freezing phenomenon familiar in the theory of GMC enters the analysis in a essential way.

The detailed regularity properties of the correlation functions are a necessary input for the proof of the Ward and BPZ identities which are based on certain exact identities involving the correlation functions and non trivial integral transforms of these correlation functions (these exact identities are obtained through Gaussian integration by parts). For instance the insertion of the SE tensor T (z) in the correlation functions leads to expressions involving Beurling and more singular integral transforms of them. The Ward and BPZ identities are then the consequence of subtle cancellations of not absolutely convergent integrals and require great care.

1.1. Perspectives. This paper is the first in a series of papers aiming at unifying two approaches of LCFT in the physics literature: the path integral approach and the conformal bootstrap approach. More precisely, the above results (Ward identities, BPZ equations, relation (1.11)) are essential in this direction.

We believe that both approaches are equivalent though the status of their relation is still controversial in the physics literature. Indeed, there are numerous reviews and papers within the physics literature on the path integral approach of LCFT and its relation with the bootstrap approach but they offer different perspectives and conclusions (see [24, 32, 37] for instance). One major step towards this unification would be the proof that C

γ

(α

1

, α

2

, α

3

) indeed satisfies the DOZZ formula, namely that one can analytically continue (α

1

, α

2

, α

3

) 7→ C

γ

(α

1

, α

2

, α

3

) to a set C

³

\ P

^γ

where P

^γ

is a (rather complicated) set of poles depending on γ and such that on C

³

\ P

γ

we have (recall that l(x) = Γ(x)/Γ(1 − x))

(1.13) C

γ

(α

1

, α

2

, α

3

) = (π µ l( γ

²

4 ) ( γ

2 )

^2−γ2/2

)

^2Q−

P i αi γ

Υ

^′γ

2

(0)Υ

^γ

2

(α

1

)Υ

^γ

2

(α

2

)Υ

^γ

2

(α

3

) Υ

^γ₂

(

^α−2Q¯ ₂

)Υ

^γ₂

(

^α−α¯₂ ¹

)Υ

^γ₂

(

^α−α¯₂ ²

)Υ

^γ₂

(

^α−α¯₂ ³

)

4In the bootstrap approach, these correlations are expressed as sums involving holomorphic (and anti holomorphic) functions and the constantsCγ(α¹, α2, α3).

5This does not mean that LCFT is a logarithmic CFT, a variant of a classical CFT.

(recall that ¯ α = α

1

+ α

2

+ α

3

) and Υ

^γ₂

is a special function (depending on γ): see subsection 7.3 for an analytic expression. The function Υ

^γ₂

has zeros but no poles and so the poles P

γ

of C

γ

(α

1

, α

2

, α

3

) can be read off formula (1.13) and correspond to the zeros of the denumerator. Assuming some regularity (in γ and α

1

, α

2

, α

3

) on C

γ

(α

1

, α

2

, α

3

) the DOZZ formula is in fact the only solution on P

γ

of shift equation (1.11) and the following dual shift equation

(1.14) C

γ

(α

1

+

_γ²

, α

2

, α

3

) C

γ

(α

1

−

γ²

, α

2

, α

3

) = − 1

π µ e

l( −

γ⁴²

)l(

^2α_γ¹

)l(

^2α_γ¹

−

γ⁴²

)l(

_γ¹

(¯ α − 2α

1

−

²γ

)) l(

¹_γ

(¯ α −

²γ

− 2Q))l(

_γ¹

(¯ α − 2α

3

−

γ²

))l(

¹_γ

(¯ α − 2α

2

−

γ²

)) . with e µ =

^(µπl(

γ2 2))

4 γ2

πl(⁴

γ2)

. Our work proves that C

γ

(α

1

, α

2

, α

3

) satisfies the first shift equation for restricted values of α

1

, α

2

, α

3

. In a forthcoming work [29], we address the problem of the analytic continuation of C

γ

(α

1

, α

2

, α

3

) as a function of α

1

, α

2

, α

3

(hence showing that one can lift the restriction on the values of α

1

, α

2

, α

3

) and the problem of proving the second shift equation (1.14).

Let us also emphasize that the functional relations (1.11) and (1.14) are very important relations per se for all γ ∈ C (and not just real γ ∈ ]0, 2])

⁶

. Indeed, for each fixed γ ∈ C , they are used in the physics literature to construct three point structure constants and CFTs in the conformal bootstrap approach: see for instance the very recent bootstrap construction of Liouville theory by Ribault-Santachiara with purely imaginary γ [35] based on the exact solution to (1.11) and (1.14) discovered in [28, 44]

⁷

.

Finally, let us mention that the Liouville three point structure constants C

γ

(α

1

, α

2

, α

3

) are also partic- ularly interesting as they seem to have deep connections to other mathematical objects, e.g. the Nekrasov partition functions: this is the basis of the celebrated AGT conjecture [1]. This brings yet additional moti- vation to study the structure constants. On the mathematical side, the AGT conjecture has been explored recently by Maulik and Okounkov in [31].

1.2. History on LCFT and probabilistic approaches to CFT. Finally we want to make some com- ments about other mathematical studies of LCFT and CFTs. There is a huge physics literature on LCFT for which we refer the reader to the reviews [43, 30, 34] and the previous references. Takhtajan et al. [40]

studied Liouville theory in the setup of a formal power series in γ (the so-called semiclassical expansion).

In particular Ward identities were established in this formal power series context. Also, independently from the work [14], Duplantier-Miller-Sheffield developped a theory of quantum surfaces in [21] which lies in some sense at the boundary of LCFT. In the case of the sphere, they work with two marked points 0 and ∞ and they consider random measures defined up to dilation and rotation since two points are not sufficient to determine conformal embeddings in the sphere. From the point view of LCFT, this corresponds to the construction of the two point correlation functions which exist only in a generalized sense: this point will be explained in more detail in the forthcoming work [29]. Their theory is based on a coupling between variants of SLE curves joining the two marked points and the full plane GFF; their framework is interesting in the study of the relation between LCFT and random planar maps.

On the constructive probabilistic side there are very few results on conformal invariance and Ward identities in other CFTs. For the Gaussian Free Field (GFF) a complete descripition of the CFT is developed in Kang and Makarov’s monograph [26]. In particular, they derive Ward and BPZ identities for this particular CFT with central charge c

GFF

= 1. Essentially, the GFF corresponds to setting the cosmological constant µ to 0. The resulting CFT has a very different structure from the Liouville case.

For interacting field theories, following the breakthrough papers by Smirnov [39] and then Chelkak- Smirnov [12], the scaling limit of the discrete Ising model in a domain at critical temperature is very well understood mathematically. The scaling limit of the model when the mesh size goes to zero is proven to be described by a unitary CFT, the Ising CFT, which belongs to the class of the so-called minimal models.

The Ising CFT is composed of two non trivial primary fields: the spin and the energy density. The rescaled correlations of the spin were proven to converge to an explicit expression in Chelkak-Hongler-Izyurov [13] (see also the independent work by Dub´edat [20] for convergence in the plane). In fact, one can also construct the

6Recall that our approach, based on a path integral, is presently restricted to the case of realγin the interval ]0,2]

7The exact solution to (1.11) and (1.14) for purely imaginaryγis not the analytic continuation inγof the DOZZ formula (1.13) valid forγ∈]0,2]

scaling limit of the spin as a random distribution: see Camia-Garban-Newman [10]. The rescaled correlations of the energy density were proven to converge to an explicit expression in Hongler-Smirnov [25] (see also the independent work by Boutillier-De Tili`ere [7, 8] for convergence in the plane). Finally, a recent work [11] shows convergence of a discrete SE tensor to a continuum limit which satisfies the associated Ward identities.

In the context of SLE and CLE, which are random conformally invariant curves, one can construct partition functions or correlation functions as probabilities of events related to several SLEs or CLE. The correlation functions correspond to CFTs with central charge c 6 1. In this context, the understanding of the correlation functions is quite precise though not complete: see Dub´edat [19], Bauer-Bernard-Kytol¨a [3]

for multiple SLEs and more recently Camia- Gandolfi-Kleban [9] for a construction of primary fields in the context of CLE. In the context of CLE, Doyon [17] has proposed a construction of the stress-energy tensor but it relies on assumptions which have not been proved yet.

1.3. Organization of the paper. The rest of the paper is organized as follows. In the next section, we set the notations, recall some results of [14] and state the main results of the paper: the Ward and BPZ identities. In Section 3, we prove results on differentiability properties of Liouville correlation functions and based on these in Section 4 we prove the Ward and BPZ identities. Finally in Section 5 we prove detailed bounds on the correlation functions when two or three points get together. This section is the technical backbone of our paper and can be seen as a first step towards proving fusion rules and operator product expansions for LCFT.

Acknowledgements. The authors wish to thank Francois David and Sylvain Ribault for fruitful discussions on Liouville Field theory. The authors also wish to thank Colin Guillarmou for helping them handle the hypergeometric equation which arises in the study of the four point correlation function.

2. Main results

2.1. Background and notations. In this section, we recall the precise definition of the Liouville correla- tion functions as given in [14].

LCFT on C ˆ . Recall that we denote by ˆ C the Riemann sphere. A smooth conformal metric on ˆ C is given by G = g(z) | dz |

²

where g(z) = O ( | z |

⁻⁴

) as z → ∞ . It is then natural to write the metric in LCFT as e

^γφ(z)

| dz |

²

= e

^γX(z)

g(z) | dz |

²

and define the Liouville action functional on ˆ C by

(2.1) S

c

(X, g) := 1 π

Z

C

| ∂

z

X(z) |

²

+ Q

c

4 g(z)R

g

(z)X(z) + πµe

^γX(z)

g(z) d

²

z.

Here R

g

(z) = − 4g

⁻¹

∂

z

∂

z¯

ln g(z) is the scalar curvature of the metric g and Q

c

=

²_γ

. Note that formally (2.1) agrees with (1.2) with φ(z) = X(z) +

_γ¹

ln g(z) up to an infinite constant which stems from the fact that C and ˆ C are not conformally equivalent. To have conformal invariance of the probabilistic theory it turns out one needs to change the parameter Q

c

=

_γ²

in the classical action (2.1) to the ”quantum” value (1.3). This yields the following definition for the quantum action

(2.2) S(X, g) := 1

π Z

C

| ∂

z

X (z) |

²

+ Q

4 g(z)R

g

(z)X(z) + πµe

^γX(z)

g(z) d

²

z.

We work in this paper with the round metric ˆ

g(z) = 4 (1 + ¯ zz)

²

in which case the scalar curvature is constant R

ˆg

= 2. The smooth metrics on ˆ C are then given by

e

^ϕ(z)

ˆ g(z) | dz |

²

with ϕ(z) and ϕ(1/z) smooth on C . In [14] it was shown that the change of the Liouville

correlation functions under change of ϕ is explicit through the so-called Weyl anomaly formula and thus

there is no loss working with ˆ g. In the sequel, we will therefore work in the background metric ˆ g.

Convention and notations. In what follows, in addition to the complex variable z, we will also consider variables x, y in C and for integer N > 3 variables z

1

, · · · , z

N

which also belong to C . For these complex variables, we will use the standard notation for complex derivatives: for x = x

1

+ix

2

we set ∂

x

=

¹₂

(∂

x1

− i∂

x2

) and ∂

x¯

=

¹₂

(∂

x1

+ i∂

x2

). The variables x, y will typically be variables of integration: we will denote by d

²

x and d

²

y the corresponding Lebesgue measure on C (seen as R

²

). We will also denote | · | the norm in C of the standard Euclidean (flat) metric and for all r > 0 we will denote by B(x, r) the Euclidean ball of center x and radius r.

Gaussian Free Field. To define the measure (1.1) it is natural to start with the quadratic part of the action functional (2.1) which naturally gives rise to a Gaussian measure, the Gaussian Free Field (GFF) (we refer to section 4 in [18] or [38] for an introduction to the GFF). As is well known the GFF on the plane is defined modulo a constant but in LCFT this constant has to be included as an integration variable in the measure (1.1). The way to proceed is to replace X in (2.1) by c + X where c ∈ R and X is the Gaussian Free Field on C where the additive constant is fixed by R

C

X (x)ˆ g(x)d

²

x = 0. The covariance of X

⁸

is given explicitely for x, y ∈ C by

(2.3) E[X (x)X(y)] = G(x, y) = ln 1

| x − y | − 1

4 (ln ˆ g(x) + ln ˆ g(y)) + χ where χ := ln 2 −

¹2

.

Gaussian multiplicative chaos. The field X is distribution valued and to define its exponential a renormaliza- tion procedure is needed. We will work with a mollified regularization of the GFF, namely X

ǫ

= ρ

ǫ

∗ X with ρ

ǫ

(x) =

_ǫ¹2

ρ(

^xx¯_ǫ2

) where ρ is C

^∞

non-negative with compact support in [0, ∞ [ and such that π R

∞

0

ρ(t)dt = 1.

The variance of X

ǫ

(x) satisfies

(2.4) lim

ǫ→0

(E[X

ǫ

(x)

²

] + ln(aǫ)) = −

¹₂

ln ˆ g(x)

uniformly on C where the constant a depends on the regularization function ρ. Define the measure

⁹

(2.5) M

γ,ǫ

(d

²

x) := e

^γ

2

2 χ

e

^{γXǫ(x)−γ}

2

2 E[Xǫ(x)²]

ˆ g(x)d

²

x.

Then, for γ ∈ [0, 2[, we have the convergence in probability

(2.6) M

γ

= lim

ǫ→0

M

γ,ǫ

and convergence is in the sense of weak convergence of measures. This limiting measure is non trivial and is a (up to a multiplicative constant) Gaussian multiplicative chaos (GMC) of the field X with respect to the measure ˆ g(x)dx (see [5] for an elementary approach and references). By (2.4) we may also write

(2.7) M

γ

(d

²

x) = lim

ǫ→0

(Aǫ)

^γ

2

2

e

^γφǫ(x)

d

²

x where the constant A = ae

^χ

, φ

ǫ

= ρ

ǫ

∗ φ and φ is the Liouville field

(2.8) φ = X +

^Q₂

ln ˆ g.

¹⁰

Note that Q here is given by (1.3) and not Q

c

. The change is due to the renormalization in (2.5).

8The fieldXwas denotedXˆg in the article [14] or the lecture notes [36].

9This normalization is chosen to match with the standard physics literature.

10This is a slight abuse of terminology as we use a slightly different convention than the article [14] or the lecture notes [36] where the Liouville field is ratherφ+c.

Liouville measure. The Liouville measure e

^−S(X,ˆg)

DX with S given by (2.2) is now defined as follows. Since R

gˆ

= 2 and R

C

X(x)ˆ g(x)d

²

x = 0 the linear term becomes R

C

(c + X (x))ˆ g(x)d

²

x = 4πc. This leads to the definition

(2.9) ν(dX, dc) := e

^−2Qc

e

^{−µeγcMγ(C)}

P(dX) dc.

Here P(dX) is the measure on H

⁻¹

( ˆ C ) induced by the GFF X . Note that the random variable M

γ

( C ) is almost surely finite because EM

γ

( C ) = e

^γ

2 2χ

R

C

ˆ g(x)dx < ∞ . This implies that ν is an infinite measure on H

⁻¹

( ˆ C ): R

dν = ∞ since for M

γ

( C ) < ∞ the c-integral diverges at −∞ . However, suitable correlation functions still exist as we see next.

Liouville correlation functions. The vertex operators (1.5) need to be regularized as well: we set (2.10) V

α,ǫ

(z) = (Aǫ)

^α22

e

^{α(φǫ(z)+c)}

= e

^αc

e

^α

2χ

2

e

^{αXǫ(z)−α22E[Xǫ(z)2]}

ˆ g

ǫ

(z)

^∆α

where ˆ g

ǫ

(z) = e

^{(ρǫ∗ln ˆg)(z)}

. Let us denote averages with respect to ν by h · i and (2.11) U

N

= { z = (z

1

, . . . , z

N

) ∈ C

^N

, z

i

6 = z

j

∀ i 6 = j } .

Fix z ∈ U

N

and weights α

1

, . . . , α

N

. Here and below we use the notation h · i

^ǫ

for the regularized Liouville measure where in (2.9) we use the measure V

γ,ǫ

(x)d

²

x instead of M

γ

. Similarly, standard GMC theory ensures that V

γ,ǫ

(x)d

²

x converges in probability to M

γ

as ǫ → 0. Now, it was shown in [14] that the limit

(2.12) h

Y

N k=1

V

αk

(z

k

) i := 4e

^−2χQ2

lim

ǫ→0

h Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ11

exists and is finite if and only if P

N

k=1

α

k

> 2Q. Moreover, under this condition, the limit is non zero if and only if α

k

< Q for all k. These conditions are the Seiberg bounds (1.6) originally introduced in [37]. We assume throughout this paper that the Seiberg bounds (1.6) are satisfied and that α

k

6 = 0 for all k.

¹²

Note that these bounds imply that for a nontrivial correlation we need at least three vertex operators; therefore, we have N > 3 in the sequel. The correlation function (2.12) satisfies the conformal invariance property (1.4) and the conformal weights of the vertex operators are given by

∆

α

= α 2 (Q − α

2 ).

Reduction to Multiplicative Chaos. In order to keep this work as self contained as possible, we remind the basics of the construction of the Liouville correlations. The main idea is that one can express these correlations as functions of GMC measures with log singularities. More precisely, using the explicit expression (2.10), we may first integrate over the c-variable in the vertex operator correlation functions which yields (see [14]):

h Y

N k=1

V

αk

(z

k

) i = 4 e

^−2χQ2

lim

ǫ→0

Z

R

e

^−2Qc

E

"

_N

Y

k=1

V

αk,ǫ

(z

k

)e

^{−µeγcRCVγ,ǫ(x)d2x}

# dc

= µ

^−s

4 e

^{χ2PNk=1α2k−2χQ2}

γ

⁻¹

Γ(s) lim

ǫ→0

E

"

_N

Y

k=1

e

^{αkXǫ(zk)−}

α2 k

2 EXǫ(zk)²

ˆ g

ǫ

(z

k

)

^∆αk

Z

C

V

γ,ǫ

(x)d

²

x

−s

#

. Now, using Girsanov’s theorem (see [14]) we may trade the vertex operators to a shift of X to obtain

(2.13) h

Y

N k=1

V

αk

(z

k

) i = 4e

^{χ2PNk=1α2k−2χQ2}

e

^{12PNi6=jαiαjG(zi,zj)}

( Y

N k=1

ˆ

g(z

k

)

^∆αk

)µ

^−s

γ

⁻¹

Γ(s)E

"Z

C

e

^{PNk=1αkG(x,zk)}

M

γ

(d

²

x)

−s

#

11The global constant 4e^−2χQ2 which depends onγplays no role but it is included to match with the standard physics literature which is based on the DOZZ formula (1.13). This constant was not included in the definitions in the article [14] or the lecture notes [36].

12This is no restriction since the caseαk= 0 corresponds to settingVα_k(zk) = 1.

where s =

PN

k=1αk−2Q

γ

. Note that the Seiberg bounds ensure s > 0. Using formula (2.3) we can write this as

(2.14) h

Y

N k=1

V

αk

(z

k

) i = B(α α α) Y

i<j

1

| z

i

− z

j

|

^αiαj

µ

^−s

γ

⁻¹

Γ(s)E

"Z

C

F(x, z)M

γ

(d

²

x)

−s

#

where we set α α α = (α

1

, · · · , α

N

),

F (x, z ) = Y

N k=1

1

| x − z

k

|

^γαk

g(x) ˆ

^{−γ4PNk=1αk}

and

(2.15) B(α α α) = 4 e

^{−χ2(PNk=1αk−2Q)2}

.

Thus, up to explicit factors the Liouville correlations are reduced to the study of the random variable R

C

F (x, z)M

γ

(d

²

x). In particular, the Seiberg bounds α

k

< Q for all k are the condition of integrability of F against the chaos measure M

γ

(see [14]).

2.2. Ward and BPZ identities. A formal calculation using the definition (1.7) of the SE tensor yields the following result in the case of LCFT

(2.16) T (z) = Q∂

²_z

φ(z) − (∂

z

φ(z))

²

+ E((∂

z

X (z))

²

).

where φ is the Liouville field (2.8). We will define this via a regularized version:

(2.17) T

ǫ

(z) := Q∂

_z²

φ

ǫ

(z) − (∂

z

φ

ǫ

(z))

²

+ E((∂

z

X

ǫ

(z))

²

) (note that X

ǫ

is smooth (a.s.)). Here is the main theorem on the Ward identities:

Theorem 2.1. (a) The correlation functions (z

1

, . . . , z

N

) 7→ h Q

N

k=1

V

αk

(z

k

) i are C

²

in the set U

N

. (b) The limits

ǫ→0

lim h T

ǫ

(z) Y

N k=1

V

αk

(z

k

) i := h T (z) Y

N k=1

V

αk

(z

k

) i (2.18)

ǫ

lim

^′→0

lim

ǫ→0

h T

ǫ

(z)T

ǫ′

(z

^′

) Y

N k=1

V

αk

(z

k

) i := h T (z)T (z

^′

) Y

N k=1

V

αk

(z

k

) i (2.19)

exist for all distinct z, z

^′

, z

1

, · · · , z

N

. (2.18) is given by the first Ward identity (1.8) and the second Ward identity (1.9) holds in the form

h T (z)T (z

^′

) Y

N k=1

V

αk

(z

k

) i = 1 2

c

L

(z − z

^′

)

⁴

h Y

N k=1

V

αk

(z

k

) i + 2

(z − z

^′

)

²

h T (z

^′

) Y

N k=1

V

αk

(z

k

) i + 1

z − z

^′

∂

z′

h T (z

^′

) Y

N k=1

V

αk

(z

k

) i

+ X

N i=1

∆

αi

(z − z

i

)

²

h T (z

^′

) Y

N k=1

V

αk

(z

k

) i + X

N i=1

1

z − z

i

∂

zi

h T (z

^′

) Y

N k=1

V

αk

(z

k

) i (2.20)

where c

L

= 1 + 6Q

²

is the central charge of the Liouville theory.

The result on the BPZ equations is the following:

Theorem 2.2. Let χ = −

^γ2

or χ = −

γ²

and suppose χ + P

N

k=1

α

k

> 2Q. Then the BPZ equation (1.10)

holds in C \ { z

1

, · · · , z

N

} .

2.3. Relations on the 3 point structure constant. We will use the BPZ equations to deduce an exact expression for the 4 point correlation function with a degenerate field h V

−^γ₂

(z) Q

3

k=1

V

αk

(z

k

) i and deduce from this expression a non trivial relation on the 3 point structure constants. This relation is usually referred to as Teschner’s trick in the physics literature since it was shown by Teschner to lead to a simple heuristic derivation of the celebrated DOZZ formula for the 3 point structure constants (see section 1.1 for the importance of these constants).

Let us first use M¨ obius invariance (1.4) to simplify the three and four point functions. This fixes the three point function up to a constant

h Y

3 k=1

V

αk

(z

k

) i = | z

1

− z

2

|

^2∆12

| z

2

− z

3

|

^2∆23

| z

1

− z

3

|

^2∆13

C

γ

(α

1

, α

2

, α

3

) (2.21)

where we denoted ∆

12

= ∆

α3

− ∆

α1

− ∆

α2

etc... Similarly, the four point function is fixed up to a single function depending on the cross ratio of the points. Specializing to the case we are interested in, we get

h V

₋^γ

2

(z) Y

3 k=1

V

αk

(z

k

) i = | z

3

− z |

^{−4∆−γ2}

| z

2

− z

1

|

^{2(∆3−∆2−∆1−∆−γ2)}

| z

3

− z

1

|

^{2(∆2+∆−γ2−∆3−∆1)}

(2.22)

× | z

3

− z

2

|

^{2(∆1+∆−γ2−∆3−∆2)}

G

(z − z

1

)(z

2

− z

3

) (z − z

3

)(z

2

− z

1

)

(2.23)

where ∆

−^γ₂

= −

^γ4

(Q +

^γ₄

) and ∆

k

=

^α₂^k

(Q −

^α2^k

). We can recover C

γ

and G(z) as the following limits C

γ

(α

1

, α

2

, α

3

) = lim

z3→∞

| z

3

|

^4∆3

h V

α1

(0)V

α2

(1)V

α3

(z

3

) i (2.24)

G(z) = lim

z3→∞

| z

3

|

^4∆3

h V

−^γ₂

(z)V

α1

(0)V

α2

(1)V

α3

(z

3

) i . (2.25)

In order to state the result, we introduce

(2.26) F

−

(z) =

2

F

1

(a, b, c, z), F

+

(z) = z

^1−c2

F

1

(1 + a − c, 1 + b − c, 2 − c, z)

where

2

F

1

(a, b, c, z) are the standard hypergeometric series extended to C \ ]1, ∞ [ and the real parameters a, b, c have the following expression

(2.27) a = γ

2 ( α

1

2 − Q 2 ) + γ

2 ( α

2

2 + α

3

2 − γ 2 ) − 1

2 b = γ 2 ( α

1

2 − Q 2 ) + γ

2 ( α

2

2 − α

3

2 ) + 1 2 and

(2.28) c = 1 + γ

2 (α

1

− Q).

As always we assume the Seiberg bounds for ( −

^γ2

, α

1

, α

2

, α

3

), i.e. P

3

k=1

α

k

> 2Q +

^γ₂

and α

k

< Q for all k.

Then:

Theorem 2.3. Let Q −

¹γ

< α

1

< Q −

^γ2

and write G(z) = | z |

^γα21

| z − 1 |

^γα22

G(z). Then e G(z) = e C

γ

(α

1

− γ

2 , α

2

, α

3

) | F

−

(z) |

²

− µ π

l( −

^γ₄²

)l(

^γα₂¹

)l(2 +

^γ₄²

−

^γα₂¹

) C

γ

(α

1

+ γ

2 , α

2

, α

3

) | F

+

(z) |

²

where l(x) =

_Γ(1−x)^Γ(x)

.

Remark 2.4. The set of (α

1

, α

2

, α

3

) that satisfy the assumptions of Theorem 2.3 is non empty for γ < √ 2.

The lower bound on α

1

can be relaxed with more work hence leading to a larger set of γ ensuring that the assumptions of Theorem 2.3 are non empty. Since this adds additional technicalities we will adress it in a forthcoming work [29]. The upper bound comes from the necessity of α

1

+

^γ₂

to satisfy the Seiberg bound α

1

+

^γ₂

< Q: one can also relax this condition to some extent with additional work [29].

From this we can deduce the following corollary on the 3 point structure constants:

Corollary 2.5. Let (α

1

, α

2

, α

3

) be such that we have Q −

¹γ

< α

1

< Q −

^γ2

and such that ( −

^γ2

, α

1

, α

2

, α

3

) satisfy the Seiberg bounds. Then we have the following relation

(2.29) C

γ

(α

1

+

^γ₂

, α

2

, α

3

) C

γ

(α

1

−

^γ2

, α

2

, α

3

) = − 1

πµ

l( −

^γ₄²

)l(

^γα₂¹

)l(

^α₂^1γ

−

^γ₄²

)l(

^γ₄

(¯ α − 2α

1

−

^γ₂

)) l(

^γ₄

(¯ α −

^γ2

− 2Q))l(

^γ₄

(¯ α − 2α

3

−

^γ2

))l(

^γ₄

(¯ α − 2α

2

−

^γ2

)) where α ¯ = α

1

+ α

2

+ α

3

.

3. Properties of Liouville Correlation Functions

3.1. Integrability Properties. In Section 5, we prove detailed estimates for the Liouville correlations as some of the points get together. For the proof of the Ward identities we need the following special cases.

Let the weights α

1

, . . . , α

N

satisfy the Seiberg bounds (1.6) and z = (z

1

, . . . , z

N

) ∈ U

N

. From now on we use the notation G

ǫ

( z ) = h Q

N

k=1

V

αk,ǫ

(z

k

) i

^ǫ

and more generally for x = x

1

, · · · , x

n

∈ C

(3.1) G

ǫ

(x; z) := h

Y

n i=1

V

γ,ǫ

(x

i

) Y

N k=1

V

αk,ǫ

(z

k

) i

^ǫ

.

This is a slight abuse of notation since we use the same notation G

ǫ

for different functions: these functions depend on the number of variables n but it should be clear from the context how many variables we are considering. Similarly G(z) and G(x; z) stand respectively for the limits of G

ǫ

(z) and G

ǫ

(x; z) as ǫ goes to 0. In the following we fix z and study G

ǫ

(x; z) as a function of x. We will mainly be interested in the case of G

ǫ

( · ; z) as a function of one variable that we will denote x or of two variables that we will denote x, y.

Define also

G(x; ¯ z) := sup

ǫ

G

ǫ

(x; z), G(x, y; ¯ z) := sup

ǫ

G

ǫ

(x, y; z)

Proposition 3.1. a) Let ǫ > 0. Then G

ǫ

(x; z) and G

ǫ

(x, y; z) are smooth with k G

ǫ

( · ; z) k

^∞

< ∞ and for some p > 2 and some constant C

ǫ

> 0

G

ǫ

(x, y; z) 6 C

ǫ

(1 + | x | )

^−p

(1 + | y | )

^−p

.

(b) The functions G(x; ¯ z ) and G(x, y; ¯ z ) belong to L

^p

( C ) and L

^p

( C

²

) respectively for p ∈ [1, r] for some r > 1 and uniformly in z on compact subsets of U

N

. The same holds for the functions

| ln | x − z

i

||

^k

G(x; ¯ z), | ln | x − y ||

^k

G(x, y; ¯ z) for all i and k > 0.

(c) Let x, y ∈ C \ { z

1

, . . . , z

N

} . Then

G(x, y; ¯ z) 6 C | x − y |

^−2+ζ

where the constant C depends on | x − z

i

| , | y − z

i

| , i = 1 . . . N.

We will also need the fact that the regularized correlations are translation invariant Lemma 3.2. For all y ∈ C , G

ǫ

(z

1

+ y, . . . , z

N

+ y) = G

ǫ

( z ) and thus P

N

i=1

∂

zi

G

ǫ

( z ) = 0.

For the proof see Section 6.1.

3.2. Integration by Parts Formula and a KPZ identity. Let f ∈ C

₀^∞

( C ) and set X (f ) = R

C

X(z)f (z)d

²

z.

The following identity follows by integration by parts in the Gaussian measure

¹³

: h X (f )

Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ

= X

N i=1

α

i

E(X (f )X

ǫ

(z

i

)) h Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ

− µγ Z

C

E(X (f )X

ǫ

(x)) h V

γ,ǫ

(x) Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ

d

²

x.

(3.2)

13Recall that for a centered Gaussian vector (X, Y¹, . . . , YN) and a smooth functionf onR^N, the Gaussian integration by parts yieldsE[X f(Y¹, . . . , YN)] =PN

k=1E[XYk]E[∂Y_kf(Y¹, . . . , YN)].

By Proposition 3.1, the x integral converges since |E(X (f )X

ǫ

(x)) | 6 C ln(2 + | x | ). We will need to use this formula for f satisfying R

C

f (z)d

²

z = 0. Then

E(X(f )X

ǫ

(z)) = (C

ǫ

∗ f )(z) −

¹₄

Z

C

ln ˆ g(u)f (u)d

²

u

where C

ǫ

= ρ

ǫ

∗ ln | z |

⁻¹

. Recalling the definition of the Liouville field (2.8) we then get h φ(f )

Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ

= X

N k=1

α

k

(C

ǫ

∗ f )(z

k

)G

ǫ

(z) − µγ Z

C

(C

ǫ

∗ f )(x)G

ǫ

(x; z)d

²

x

+ 1 4

Z

C

ln ˆ g(u)f (u)d

²

u (2Q − X

N k=1

α

k

)G

ǫ

(z) + µγ Z

C

G

ǫ

(x; z)d

²

x The metric dependent term actually vanishes due to the following remarkable identity

Lemma 3.3. For all ǫ > 0 (KPZ-identity)

(3.3) µγ

Z

C

G

ǫ

(x; z)d

²

x = ( X

N k=1

α

k

− 2Q)G

ǫ

(z).

Proof. Recalling the c-dependence in (2.10) we get by a simple change of variables γ

⁻¹

ln µ + c = c

^′

that h

Y

N k=1

V

αk,ǫ

(z

k

) i

^ǫ,µ

= Z

R

e

^−2Qc

E

"

_N

Y

k=1

V

αk,ǫ

(z

k

)e

^{−µRCVγ,ǫ(x)d2x}

# dc

=µ

⁻

PN k=1αk−2Q

γ

Z

R

e

^−2Qc′

E

"

_N

Y

k=1

V

αk,ǫ

(z

k

)e

^{−RCVγ,ǫ(x)d2x}

# dc

^′

.

The identity follows by differentiating in µ. The limit as ǫ → 0 follows in virtue of Proposition 3.1.

Now, we give some definitions which will be used throughout the rest of the paper. Recall that ρ

ǫ

(x) =

1

ǫ²

ρ(

^¯xx_ǫ2

) where ρ is C

^∞

non-negative with compact support in [0, ∞ [ and such that π R

∞

0

ρ(t)dt = 1.Then we set for ǫ, ǫ

^′

> 0

(3.4) 1

(z)

_ǫ,ǫ′

:= ρ

ǫ

∗ ρ

ǫ^′

∗ 1 z and more generally for n > 0

(3.5) 1

(z)

ⁿ⁺¹_ǫ,ǫ′

:= ∂

_zⁿ

1 (z)

_ǫ,ǫ′

If ǫ

^′

= 0, we simply set

_(z)¹

ǫ

:=

_(z)¹

ǫ,0

and similarly for

_(z)¹n+1 ǫ

.

We then get the integration by parts formula that will be used repeatedly in this paper:

Lemma 3.4. For all ǫ > 0 and ǫ

^′

> 0 and n > 0 h ∂

_zⁿ⁺¹

φ

ǫ^′

(z)

Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ

= − 1 2

X

N i=1

α

i

∂

_zⁿ

1 (z − z

i

)

_ǫ,ǫ′

G

ǫ

(z) + µγ 2

Z

C

∂

_zⁿ

1 (z − x)

_ǫ,ǫ′

G

ǫ

(x; z)d

²

x.

(3.6)

As an application, let us compute the derivative of the regularized correlations:

(3.7) ∂

zi

G

ǫ

(z) = α

i

h ∂

zi

φ

ǫ

(z

i

) Y

N k=1

V

αk,ǫ

(z

k

) i

ǫ

= − 1 2

X

N j;j6=i

α

i

α

j

(z

i

− z

j

)

ǫ,ǫ

G

ǫ

(z) + α

i

µγ 2 Y

i,ǫ

(z) with

(3.8) Y

i,ǫ

(z) =

Z

C

1 (z

i

− x)

ǫ,ǫ

G

ǫ

(x; z)d

²

x.

Remark. Note that Proposition 3.1 does not allow us to control the limit as ǫ tends to zero of expression (3.8) since it guarantees only integrability of a logarithmic singularity in z

i

− x. Indeed, depending on γ and α

i

the singularity of correlation function G

ǫ

(x; z) is | z

i

− x |

^−2+ζ

with ζ arbitrarily close to 0. Hence the integral in (3.8) is not always absolutely convergent and its analysis is rather subtle.

3.3. Differentiability of Correlation Functions. This Section is devoted to the proof of Theorem 2.1 (a). We start with a convenient representation of the first derivative ∂

zi

G

ǫ

(z); indeed, as mentioned in the previous remark, the Y

i,ǫ

(z) term in (3.7) is difficult to study directly hence we give an expression for the Y

i,ǫ

(z) term which is easier to study:

Lemma 3.5. We can write ∂

zi

G

ǫ

( z ) under the following form (3.9) ∂

zi

G

ǫ

(z) = − 1

2 X

N j;j6=i

α

i

α

j

(z

i

− z

j

)

ǫ,ǫ

G

ǫ

(z) + Z

C

f

i,ǫ

(x, z)G

ǫ

(x; z)dx + Z

C

Z

C

F

ǫ

(x, y, z

i

)G

ǫ

(x, y; z)d

²

xd

²

y where f

i,ǫ

and F

ǫ

are bounded smooth functions. Furthermore, for all j, f

i,ǫ

, F

ǫ

, ∂

zj

f

i,ǫ

, ∂

zj

F

ǫ

, ∂

x

f

i,ǫ

and (∂

x

+ ∂

y

)F

ǫ

converge uniformly as ǫ → 0 to bounded continuous limits. Moreover

(3.10) k ∂

ⁿ

f

i,ǫ

k

∞

, k ∂

ⁿ

F

ǫ

k

∞

6 Cδ

⁻ⁿ⁻¹_i

n = 0, 1

uniformly in ǫ where δ

i

:= min

j;j6=i

| z

i

− z

j

| and C is some global constant (which depends only on µ, γ and α

i

). In (3.10) ∂ can be taken as ∂

zj

, ∂

x

(for f

i,ǫ

) or ∂

x

+ ∂

y

(for F

ǫ

).

Proof. The key idea behind this proof is to use an exact identity (i.e. identity (3.11) below) to deduce a new and easier to study expression for Y

i,ǫ

(z), which we will then plug into (3.7) to prove our claim.

To obtain this new expression for Y

i,ǫ

(z), recall that x 7→ G

ǫ

(x; z) is smooth. Hence (π times) the Beurling transform

A

ǫ

(z; z) = − Z

C

1

(z − x)

²

G

ǫ

(x; z)d

²

x := − lim

ǫ^′→0

Z

C

1

(z − x)

²

1

|z−x|>ǫ^′

G

ǫ

(x; z)d

²

x is defined and satisfies the following key identity

(3.11) A

ǫ

(z; z ) =

Z

C

1

z − x ∂

x

G

ǫ

(x; z )d

²

x = B

ǫ

(z; z ) + C

ǫ

(z; z ) where by (3.7)

B

ǫ

(z; z ) = γ 2

X

N j=1

α

j

Z

C

1 z − x

1

(z

j

− x)

ǫ,ǫ

G

ǫ

(x; z ) dx C

ǫ

(z; z) = − µγ

²

2 Z

C

1 z − x

Z

C

1 (x − y)

ǫ,ǫ

G

ǫ

(x, y; z)d

²

xd

²

y.

Using

_z−x¹

=

_z−z¹_j

_x−z

j

z−x

+ 1

we write (3.12)

Z

C

1 z − x

1 (z

j

− x)

ǫ,ǫ

G

ǫ

(x; z)dx = 1 z − z

j

(Y

j,ǫ

(z) + D

j,ǫ

(z; z)) where

D

j,ǫ

(z; z) = − Z

C

z

j

− x z − x

1 (z

j

− x)

ǫ,ǫ

G

ǫ

(x; z) d

²

x.

We want to deduce from expression (3.11) (valid if z 6 = z

j

for all j) an expression for Y

i,ǫ

(z). A natural

way to do so is to integrate expression (3.11) with respect to a small contour around z

i

to get rid of the

Y

j,ǫ

(z) terms for j 6 = i. However, contour integrals lack smoothness so we will use an equivalent but more

smooth procedure which boils down to taking an average over contour integrals of (3.11). More precisely, let

θ(x) be a smooth bump with support on ]

^δ₄ⁱ

,

^δ₂ⁱ

[ and χ(z) = θ( | z | )

_|z|^z

. So χ is supported inside the annulus